Our main theoretical result is shown in Theorem 1, which claims a convergence rate of \(\mathcal {O}(\sqrt{n} K^{-2/5})\)  for achieving an approximate stationary and optimal solutions with respect to variables \(x\)  and \(y,\lambda \) . Overall, our analysis follows a similar overall structure as the analysis in [1]}. However, we highlight a few unique differences due to the task-robust nature of our formulation and algorithm. First, as in TTSA, the convergence of \(x\) 's relies on the \(y\)  being close to \(y^*(x)\) . However, we are trying to fit one \(x\)  across multiple functions \(f_i\) . In our analysis, therefore, convergence is limited by the worst \(y_i\) , i.e. \(\max _{i \in [n]} \Vert y_i - y_i^{\star }(x)\Vert \) . Second, since objective values are weighted by \(\lambda \) , it may still be small for suboptimal \((x, y)\)  if \(\lambda \)  does not converge. Therefore, we also require a convergence result for \(\lambda \) . To do so, we take a similar approach by leveraging the theoretical result in [2]}.
